Transient solutions of Markov processes by Krylov subspaces

In this note we exploit the knowledge embodied in infinitesimal generators of Markov processes to compute efficiently and economically the transient solution of continuous time Markov processes. We consider the Krylov subspace approximation method which has been analysed by Y. Saad for solving linear differential equations. We place special emphasis on error bounds and stepsize control. We discuss the computation of the exponential of the Hessenberg matrix involved in the approximation and an economic evaluation of the Pade method is presented. We illustrate the usefulness of the approach by providing some application examples